Question 373688
A Norman window has the shape of a semicircle atop a rectangle so that the
 diameter of the semicircle is equal to the width of the rectangle.
 The goal is to find the area of the largest possible Norman window with a
 perimeter of 35 feet? 
:
Let x = the width and diameter of the window
Let L = the rectangle portion length of the window
:
radius of the semicircle = .5x
half circumference of the semi circle = {{{pi*.5x}}} = 1.5708x
:
Perimeter:
1.5708x + x + 2L = 35
2.5708x + 2L = 35
2L = (35-2.5708x)
L = {{{1/2}}}(35-2.5708x)
L = 17.5 - 1.2854x
:
The area:
A = x * L
Replace L with (17.5-1.2854x)
A = x(17.5-1.2854x)
A = -1.2854x^2 - 17.5x
Max area occurs at the axis of symmetry of this quadratic equation
x = {{{(-17.5)/(2*-1.2854)}}}
x = {{{(-17.5)/(-2.5708)}}}
x = 6.8072 ft the width that gives max area
:
Find the area, replace x with 6.8072
A = -1.2854(6.8072^2) + 17.5(6.8072)
A = -1.2854(46.3382) + 17.5(6.8072)
A = -59.556 + 119.126
A = 59.57 sq/ft, max area for 35 ft perimeter